Group rings of finite strongly monomial groups: central units and primitive idempotents

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We compute the rank of the group of central units in the integral group ring G of a finite strongly monomial group G. The formula obtained is in terms of the strong Shoda pairs of G. Next we construct a virtual basis of the group of central units of G for a class of groups G properly contained in the finite strongly monomial groups. Furthermore, for another class of groups G inside the finite strongly monomial groups, we give an explicit construction of a complete set of orthogonal primitive idempotents of G. Finally, we apply these results to describe finitely many generators of a subgroup of finite index in the group of units of G, this for metacyclic groups G of the form G=Cqₘ Cₚₙ with p and q different primes and the cyclic group Cₚₙ of order pⁿ acting faithfully on the cyclic group Cqₘ of order qᵐ.